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Abstract

Groups of satellites flying in formation require maintaining the specific relative geometry of the formation with high precision. This requirement implies to consider the problem of relative station keeping in a renewed framework. In this framework, issues related to the derivation of reliable relative models as well as to the peculiarity of the synthesis problems must be jointly considered. This paper presents some preliminary results of a robust multi- objective control approach applied to the station keeping of a low Earth observation system, i.e. the interferometric cartwheel, patented by CNES. This wheel is made up of three receiving spacecrafts, which follow an emitting Earth observation radar satellite. The particular geometry of this formation of satellites leads to the derivation of a simplified uncertain state-space model. Atmospheric drag perturbations are included in the linearized equations of the relative motion and the atmospheric density part of the definition of the atmospheric drag force is considered to be uncertain due to its dependence upon the solar activity. In the first part of the paper, an uncertain polytopic state-space model is derived. The second part describes the station keeping strategy of the formation. The station keeping strategy is performed using pure passive actuators. Due to the high stability of the relative eccentricity of the formation, only the relative semi major axis has to be controlled. Differential drag due to a differential orientation of the solar panel is used here to control relative altitude. A robust multi-objective control strategy via state-feedback is developed and tested as autonomous orbit controller. These results are analyzed via highly non linear simulations performed on a platform of CNES.

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Abstract

Recent papers in the field of LMI-based robust control have provided extensions of known results for linear time-invariant systems to the case of periodically time varying linear systems. These results, theoretically satisfactory because formulated in terms of optimization problems of polynomial complexity, may still have limited applications in practice because the number of variables and constraints is very large. The present paper proposes a new formulation of these results that allows to reduce the computational burden both by reducing the number of decision variables and the size of the constraints. Along with this numerical improvement, the paper produces a new modeling of periodic discrete-time systems in descriptor form that is believed promising for future research.

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In this paper, we propose a new LMI-based method for robust state-feedback controller synthesis of discrete-time linear periodic/time-invariant systems subject to polytopic uncertainties. In stark contrast with existing approaches that are confined to static controller synthesis, we explore dynamic controller synthesis and reveal a particular periodically timevarying dynamical controller structure that allows LMI-based synthesis. In particular, we prove rigorously that the proposed design method encompasses the well-known extended-LMI-based design methods as particular cases. Through numerical experiments, we demonstrate that the suggested design method is indeed effective to achieve less conservative results.

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We apply robust control technics to an adaptive optics system including a dynamic model of the deformable mirror. The dynamic model of the mirror is a modification of the usual plate equation. We propose also a state-space approach to model the turbulent phase. A continuous time control of our model is suggested taking into account the frequential behavior of the turbulent phase. An H1 controller is designed in an infinite dimensional setting. Due to the multivariable nature of the control problem involved in adaptive optics systems, a significant improvement is obtained with respect to traditional single input single output methods.

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The problem of the stationkeeping for a small spacecraft is studied and a solution based on periodic feedback control laws is considered. Linearized equations of the relative motion of the satellite near an eccentric reference orbit are derived in the presence of the second zonal gravitational harmonic J2 and atmospheric drag perturbations. The obtained linear continuous-time model of the relative motion is Tperiodic where T is the orbital period. After a discretization of the model, a state-feedback control law with performance requirement defined by the generalized H2 operator norm may be computed by a linear matrix inequality-based algorithm. Illustrative nonlinear simulations show the efficiency of the approach based on the use of linearized spacecraft relative motion dynamics associated to systematic H2 synthesis of stabilizing memoryless N-periodic state-feedback control laws.